The characters:
- 3-dimensional manifolds and knots
- 4-dimensional manifolds and knotted surfaces
Why?
How?
- Measuring knottedness of surfaces via the length of a regular homotopy
2022-06-07
The characters:
Why?
How?
Every classical knot \(k \colon \mathbb{S}^{1} \hookrightarrow \mathbb{S}^{3}\) is homotopic to the unknot.
Generically, the homotopy \(H \colon \mathbb{S}^{1} \times [0, 1] \rightarrow \mathbb{S}^{3}\) is a sequence of isotopies and crossing changes.
Knowing how the (immersed) surfaces in a 4-manifold \(M^{4}\) interact can tell us a lot about the topology of \(M\)
Given non-closed 4-manifold \(X^{4}\) with 3-manifold boundary \(\partial X = Y\); classical knot \(k \colon \mathbb{S}^{3} \hookrightarrow Y^{3}\),
Related to higher dimensions:
In a 5-dimensional cobordism, the attaching spheres of the 3-handles are 2-knots in 4-manifolds.
(Smale 1958): For every smoothly knotted 2-sphere \(K \colon \mathbb{S}^{2} \hookrightarrow \mathbb{S}^{4}\), there is a regular homotopy starting at the embedding \(K\) and ending at the unknot.
This “movie of movies” is a regular homotopy from a non-trivially knotted 2-sphere (left) to the unknotted sphere in \(\mathbb{S}^{4}\) (right)
Define the Casson-Whitney number \[ \mathop{\mathrm{u_{\mathrm{CW}}}}(K) \text{ of } K \colon \mathbb{S}^{2} \hookrightarrow \mathbb{S}^{4} \] as the minimal number of finger moves in a regular homotopy \(K \leadsto\) unknot.
\(\pi K = \pi_{1}(\mathbb{S}^{4} - K) \twoheadrightarrow H_{4}(\mathbb{S}^{4} - K) \cong \mathbb{Z}\cong \langle t \rangle\)
Study the homology of the associated infinite cyclic cover:
\[ \langle t \rangle \curvearrowright \widetilde{(\mathbb{S}^{4} - K)}_{\mathbb{Z}} \rightarrow \mathbb{S}^{4} - K \]
The Alexander module is the following \(\mathbb{Z}[t, t^{1}]\)-module:
\[ \begin{align} H_{1}(\widetilde{(\mathbb{S}^{4} - K)}_{\mathbb{Z}}) & \cong [\pi K, \pi K] \left/ [[\pi K, \pi K], [\pi K, \pi K]] \right. \\ & \cong \pi K^{(1)} \left/ \pi K^{(2)} \right. \end{align} \]
\[ \pi_{1}(\mathbb{S}^{4} - K') \cong { \pi_{1}(\mathbb{S}^{4} - K) } \; \left/ \; { \langle \langle [ {\color{purple} w}^{{\color{purple} {-1}}} {\color{red} a} {\color{purple} w}, {\color{green} b}] \rangle \rangle } \right. \]
Slogan: Finger moves can make a pair of meridians commute.
\[ { \Large \begin{align} \mathop{\mathrm{u_{\mathrm{CW}}}}(K) & \ge \; \substack{\text{minimal number of finger move relations} \\ [w_{i}^{-1} a_{i} w_{i}, a_{i}] \text{ to make } \pi_{1}(\mathbb{S}^{4} - K) \text{ abelian}} \\[1em] & \ge \; \substack{\text{minimal number of relations} \\ w_{i}^{-1} a_{i} w_{i} = a_{i} \text{ to make } \pi_{1}(\mathbb{S}^{4} - K) \text{ abelian}} \\[1em] & \ge \; \substack{\text{minimal size of generating set} \\ \text{of Alexander module of } K} \end{align} } \]
Prop. \(\mathop{\mathrm{u_{\mathrm{CW}}}}\) can grow arbitrarily large:
There are 2-knots \(K_{n} \colon \mathbb{S}^{2} \hookrightarrow \mathbb{S}^{4}\) with \(\mathop{\mathrm{u_{\mathrm{CW}}}}(K_{n}) = n\).
\(K =\) 0-twist spin of \(T_{2, p} \mathop \#T_{2, q}\) is a 2-knot
\(q = p + 2\) or \(q = p + 4\) or (\(q = p + 6\) and \(\gcd(p, p+6) = 1\))
Thm. For \(K_{1}, K_{2}\) knotted surfaces in \(\mathbb{S}^{4}\) with determinant \(\neq 1\), the Casson-Whitney number of the connected sum is \(\mathop{\mathrm{u_{\mathrm{CW}}}}(K_{1} \mathop \#K_{2}) \ge 2\).
Rim surgery: surface \(F^{2} \subset X^{4}\), pattern knots \({\color{blue} J_{\color{blue} 1}}, {\color{blue} J_{\color{blue} 2}} \colon \mathbb{S}^{1} \hookrightarrow \mathbb{S}^{3}\)
(R., 2021): \(\mathop{\mathrm{length_{\textrm{CW}}}}(F_{\tau^{m}}({\color{green} \alpha}, {\color{blue} J_{\color{blue} 1}}), F_{\tau^{m}}({\color{green} \alpha}, {\color{blue} J_{\color{blue} 2}})) \le \mathop{\mathrm{dist_{\textrm{Gord}}}}({\color{blue} J_{\color{blue} 1}}, {\color{blue} J_{\color{blue} 2}})\)
Thm. The Casson-Whitney number \(\mathop{\mathrm{u_{\mathrm{CW}}}}(\tau^{n} k)\) of every twist spin of \(k \colon \mathbb{S}^{1} \hookrightarrow \mathbb{S}^{3}\) is a lower bound for the classical unknotting number of the original knot \(k\).
Special case of (Scharlemann 1985): \(k_{1}, k_{2}\) classical knots with nontrivial determinant, then the classical unknotting number \(u(k_{1} \mathop \#k_{2}) \ge 2\).
(R., 2021): \(\exists\) a family of 2-knots \(K_{N} \colon \mathbb{S}^{2} \hookrightarrow \mathbb{S}^{4}\) such that the Casson-Whitney length to any ribbon 2-knot \(\rightarrow \infty\).
Suciu’s infinite family of ribbon 2-knots \(R_{l} \colon \mathbb{S}^{2} \hookrightarrow \mathbb{S}^{4}\)
For every \(k \in \mathbb{N}\), \(\pi_{1}(\mathbb{S}^{4} - R_{k})\) is the trefoil knot group
(Kanenobu and Sumi 2020): \(R_{k} \mathop \#\mathbb{RP}^{2} \neq R_{l} \mathop \#\mathbb{RP}^{2}\) for \(k \neq l \in \mathbb{N}\)
(R., 2021): For all \(l \ge 1\) we have \(\mathop{\mathrm{length_{\textrm{CW}}}}(R_{l} \mathop \#\mathbb{RP}^{2}, \mathbb{RP}^{2}) = 1\).
Freedman, Michael, Robert Gompf, Scott Morrison, and Kevin Walker. 2010. “Man and Machine Thinking about the Smooth 4-Dimensional Poincaré Conjecture.” Quantum Topol. 1 (2): 171–208. https://doi.org/10.4171/QT/5.
Joseph, Jason M., Michael R. Klug, Benjamin M. Ruppik, and Hannah R. Schwartz. 2021. “Unknotting Numbers of 2-Spheres in the 4-Sphere.” J. Topol. 14 (4): 1321–50. https://doi.org/10.1112/topo.12209.
Kanenobu, Taizo, and Toshio Sumi. 2020. “Suciu’s Ribbon 2-Knots with Isomorphic Group.” J. Knot Theory Ramifications 29 (7): 2050053, 9. https://doi.org/10.1142/S0218216520500534.
Klug, Michael, and Benjamin Ruppik. 2021. “Deep and Shallow Slice Knots in 4-Manifolds.” Proc. Amer. Math. Soc. Ser. B 8: 204–18. https://doi.org/10.1090/bproc/89.
Manolescu, Ciprian, Marco Marengon, and Lisa Piccirillo. 2020. “Relative Genus Bounds in Indefinite Four-Manifolds.” https://arxiv.org/abs/2012.12270.
Scharlemann, Martin G. 1985. “Unknotting Number One Knots Are Prime.” Invent. Math. 82 (1): 37–55. https://doi.org/10.1007/BF01394778.
Smale, Stephen. 1958. “A Classification of Immersions of the Two-Sphere.” Trans. Amer. Math. Soc. 90: 281–90. https://doi.org/10.2307/1993205.
I found a 4-manifold \(X^{4}\) and non-local knots in \(\partial X\) which are
(Freedman et al. 2010)’s attempt at proving a proposed smooth homotopy 4-ball \(\mathcal{B}\) (= smooth contractible compact 4-manifold with \(\mathbb{S}^{3}\) boundary) is exotic:
I found an infinite family of topologically shallow slice and smoothly deep slice knots in the knot trace \(X_{-1}(T_{2, -3})\):
\((p, 1)\)-cable \(J_{p, 1}\) of the knots \[ J = \textrm{Wh}^{(3)}_{+}( \textrm{Wh}_{+}(K^{\ast}, +1), 0 ) \subset \mathbb{S}^{3}_{-1}(T_{2, -3}) = P \]
\[ J = \textrm{Wh}_{+, 1\textrm{st}}^{(3)}(\mathop{\mathrm{BD}}(K^{\ast}, 0), 0), \]
(R., 2021): There exists a 2-component link \(J\) in the boundary \(Y = \partial X\) of the 2-handlebody \(X = X_{-1}(T_{2, -3})\) with the following properties: